PHY6095/PHZ6166: Final exam
due: Tuesday, 04/30, 10 a.m.
submit your work either in person or by e-mail
You must work individually to receive full credit
2
• Problem 1: 34 points
A neutron star can be considered as an ideal and degenerate Fermi gas at T = 0. The star is held together
by a balance between the outward Pauli pressure and inward gravitational force. Assuming that the star is
spherically symmetric, the gravitational potential φ satisfies the Newton equation
∇2 φ = 4πGρ(r),
where G is the gravitational constant and ρ(r) is the neutron density. The potential energy of a neutron is
U = mn φ, where mn is the neutron mass. In equilibrium, µ(r) + mn φ(r) = const, where µ(r) is the chemical
potential of the neutron gas.
– Find the condition for the neutron gas to be non-relativistic. The condition must be in a form of strong
inequality (. . . . . .). (Hint: Write down the scaling forms of ρ and µ as a function of the distance from
the star’s center but do not attempt to find the scaling functions explicitly.)
– Assuming that the condition above is satisfied, find how the total mass of the star scales with the star’s
radius, R.
– Suppose now that the neutron gas is in the ultra-relativistic regime, i.e., the neutron energy En = pc,
where p is its momentum. Find how the the total mass of the star depends on R in this case.
• Problem 2: 33 points
– Find the asymptotic behavior of the following integral both for a 1 and a 1:
√
Z ∞
cos(a x)
.
I(a) =
dx
x+1
0
– Estimate the integral
Z
J(a) =
∞
r
dx
0
x −(x+ xa )
e
.
1+x
for a 1.
• Problem 3: 33 points
A regular hexagon has C6v symmetry. The table of characters of this group is given below. According to this
table, a polar vector (x, y, z) transforms as Γ0 = A1 ⊕ E1 .
– Explain why another 2D representation, E2 , is not a suitable representation for a polar vector.
– Which transitions can be induced by an external electric field?
C6v
z A1
A2
B1
B2
(x, y) E1
E2
I
1
1
1
1
2
2
C2 2C3
1
1
1
1
−1
1
−1
1
−2 −1
2 −1
2C6 3σv 3σv0
1 1 1
1 −1 −1
−1 −1 1
−1 1 −1
1 0 0
−1 0 0